Drivers breaking down after model transformation
Bad choice of WORLD space.
To test Added a simple driver #frame - 1 to the drive shaft to get it spinning.
As suspected the issues are caused by a
i) Using rotation transform driver type, with WORLD_SPACE

Parented the whole set up to an empty, rotating the world space of the parent changes the world space of the children and causes irregular results.
Recommend either. Lesser
Change all to local space.  Here is a script to select all parts and will change transform  type driver targets to local space.
import bpy
context = bpy.context

for ob in context.selected_objects:
    if not ob.animation_data:
        continue
    for d in ob.animation_data.drivers:
        for v in d.driver.variables:
            if v.type == 'TRANSFORMS':
                for t in v.targets:
                    t.transform_space = 'LOCAL_SPACE'

After which

Note The outer gear is driven by the stationary  hidden object, No idea why it worked at all.
However as mentioned in comments Eulers are prone to gimbal lock and since it's local could simply use the property.  (Also makes sure the rotation is not restricted to (-180. 180) which can be the case in constraints and can be difficult to diagnose if not aware)
Finally, would contend that all gears could be driven directly by the rotation of the drive shaft. For most cases a custom property &quot;drive&quot; could be added to the gear and the driver could be
self[&quot;ratio&quot;] * drive

where drive is a single property variable pointing to the shaft Y rotation, or in the case shown (enable Use Self in the driver)
self[&quot;ratio&quot;} * (frame - 1)

since I set that as the drive shaft Y rotation.